Euler Equations and Monetary Policy∗ Fabrice Collard†
Harris Dellas‡
July 2, 2007
Abstract Euler equations are the key link between monetary policy and the real economy in NK models. As is well known, Euler equations under standard separable preferences find it difficult to generate behavior that matches that implied by VARs. Allowing for time non-separability (habits) solves some of these problems (Fuhrer, 2000) but does not help with or even exacerbates other problems (Canzoneri, Cumby and Diba, 2006). We show that introducing also nonseparability between leisure and consumption significantly ameliorates these remaining problems and allow Euler equations to be a more reliable tool for the study of the effects of monetary policy. JEL class: E10, E43, E44, E52 Keywords: Euler interest rate, non-separable utility, monetary policy
∗
We would like to thank Sergio Rebelo and Mike Woodford for valuable comments. Toulouse School of Economics, Manufacture des Tabacs, bˆ at. F, 21 all´ee de Brienne, 31000 Toulouse, France. Tel: (33–5) 61–12–85–42, Fax: (33–5) 61–22–55–63, email:
[email protected], http://fabcol.free.fr ‡ Department of Economics, University of Bern, CEPR. Address: VWI, Schanzeneckstrasse 1, CH-3012 Bern, Switzerland. Tel: +41(0)31-631-3989, Fax: +41(0)31-631-3992, email:
[email protected], http://www.vwi.unibe.ch/amakro/ †
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Introduction The consumption Euler equation is a crucial component of modern macroeconomic models. It plays a particularly important role in monetary models with nominal rigidities as, in these models, the nominal interest rate is the instrument of monetary policy and a key transmission mechanism. The fact that Euler equations do not always perform satisfactorily is well known. For instance, the equity premium puzzle literature has highlighted the inability of the standard model to generate realistic behavior for the real interest rates. Nonetheless, such failures do not seem to have undermined the faith of economists on standard Euler equation formulations when modelling macroeconomic policy. What probably lies behind this is the belief that, in spite of their empirical weaknesses, the Euler equations still provide a reliable and useful framework for analyzing the conduct and effects of monetary policy. Two recent papers, one by Furher, 2000 and another by Canzoneri et al. 2006, have challenged this belief. Fuhrer, 2000, argues that, following monetary policy shocks, the standard model with time separable preferences cannot give rise to the hump-shaped responses of spending and inflation that are observed in VARs. Consequently, the model fails to capture the main transmission mechanism of monetary policy. He then proceeds to demonstrate how the introduction of time non-separability (habits) alleviates the problem of poor dynamics. Canzoneri et al., 2006, show that while habits may be useful along this dimension, they may not help for (and may even exacerbate) other problems associated with the empirical performance of Euler equations. They do so using a novel approach: They first estimate a VAR that contains consumption and inflation. From the VAR they derive the moments as well as the dynamics of consumption and inflation. They then use this information in a standard, consumption Euler equation to derive the implied paths of nominal and real interest rates. Their analysis establishes that the implied interest rates behave strikingly differently from the money market interest rates. In particular, the correlation between the rates implied by consumption Euler equations and the money market rates is strongly negative. Moreover, the Euler equation rate responds negatively to an increase in the FED rate. These findings are robust across a large number of specifications of utility, such as additively separable CRRA and various commonly used forms of habits (e.g., the external habits used by Fuhrer, 2000, the internal habits used by Christiano et al., 2005, Abel’s model of catching up with the Joneses, 1999 and Campbell and Cochrane’s model of external habit). Consequently, they seem to represent a serious blow to models, such as the New Keynesian model, where monetary policy operates through changes in the real interest rate. In order to understand the forces at hand and also to gain some insight regarding the generality of this result consider monetary tightening in the New Keynesian model. Consumption is known to respond sluggishly (see Christiano et al., 2005), so expected consumption growth declines. The Euler equation real interest rate is positively related to expected consumption growth, consequently, 2
it declines too. But monetary tightening in the real world is typically associated with an increase in the money market real rate. Hence, policy induced changes in the Euler equation real rate and the money market real rate are negatively correlated. The situation can be different under a formulation of preferences that allows employment decisions to influence the consumption decision (non-separability between consumption and leisure). In this case, the consumption Euler equation also includes the moments of employment. The relationship between the real interest rate and employment growth1 may be either positive or negative. In general, strong wealth effects (an elasticity of intertemporal substitution that exceeds unity) can make this relationship negative. If this negative relationship dominates the positive one associated with consumption, then changes in monetary policy can induce changes in real interest rates in the Euler equation that conform those in the real world. For instance, following contractionary policy both the Euler equation and the money market real rates can go up. The objective of this paper is to investigate whether allowing for non-separability between consumption and employment in the utility function renders the Euler equations a more reliable tool for the study of the effects of monetary policy. We find that this is indeed the case. In particular, non-separability makes the problems identified by Canzoneri et al., 2006, less severe, even if it does not succeed in completely solving them. In section 1 we derive the real and nominal interest rates implied by consumption Euler equations under a separable and non-separable utility functions. In section 2 we describe the VAR specification that serves as the input for the generation of the Euler equation interest rates.
1
The model
In this section we compute real and nominal interest rates implied by consumption Euler equations for a fairly general specification of the utility function. Implicit in what follows is the existence of a representative consumer who makes consumption, savings and leisure decisions in order to maximize lifetime utility function subject to a standard intertemporal budget constraint.
1.1
Preferences
The utility function is assumed to take the form ϕ ((Ct /Ct−1 )ν `t1−ν )1−σ u(Ct , `t ) = 1−σ
where Ct denotes consumption and `t = 1 − ht is leisure time. This specification features non separable consumption–leisure decisions and non–time separable consumption decisions. The pa1 Employment is known to respond sluggishly, see Christiano et al., 2005, so expected employment growth declines following monetary tightening.
3
rameter ν ∈ [0, 1] is the weight assigned to consumption in the utility function while ϕ ∈ [0, 1] controls the degree of habits2 . Finally, σ > 0 is the risk aversion parameter. The associated Euler equation is then given by h i ν(1−σ)−1 −ϕν(1−σ) (1−ν)(1−σ) ν(1−σ) −ϕν(1−σ)−1 (1−ν)(1−σ) E C C ` − βϕC C ` /πt+1 t t t+1 t+1 t+2 t+1 t+2 1 h i =β ν(1−σ)−1 −ϕν(1−σ) (1−ν)(1−σ) ν(1−σ) −ϕν(1−σ)−1 (1−ν)(1−σ) 1 + it E C C ` − βϕC C ` t
t
t−1
t
t+1
t
t+1
where β ∈ (0, 1) is the discount factor, it is the nominal interest rate and πt = Pt /Pt−1 is the gross rate of inflation.3 Assuming log–normality of consumption and inflation this rewrites as exp(x1t ) − βϕ exp(x2t ) 1 =β 1 + it exp(x3t ) − βϕ exp(x4t ) with x1t =(ν(1 − σ) − 1)Et ct+1 − ϕν(1 − σ)ct + (1 − ν)(1 − σ)Et `t+1 − Et πt+1 (ν(1 − σ) − 1)2 ((1 − ν)(1 − σ))2 Vt πt+1 Vt ct+1 + Vt `t+1 + 2 2 2 − (1 − ν)(1 − σ)Ct (`t+1 , πt+1 ) + (ν(1 − σ) − 1)(1 − ν)(1 − σ)Ct (ct+1 , `t+1 )
+
− (ν(1 − σ) − 1)Ct (ct+1 , πt+1 ) x2t =ν(1 − σ)Et ct+2 − (ϕν(1 − σ) + 1)Et ct+1 + (1 − ν)(1 − σ)Et `t+2 − Et πt+1 (ν(1 − σ))2 (ϕν(1 − σ) + 1)2 ((1 − ν)(1 − σ))2 Vt πt+1 Vt ct+2 + Vt ct+1 + Vt `t+1 + 2 2 2 2 − ν(1 − σ)Ct (ct+2 , πt+1 ) + (ϕν(1 − σ) + 1)Ct (ct+1 , πt+1 ) − (1 − ν)(1 − σ)Ct (`t+2 , πt+1 )
+
− ν(1 − σ)(ϕν(1 − σ) + 1)Ct (ct+2 , ct+1 ) + ν(1 − ν)(1 − σ)2 Ct (ct+2 , `t+2 ) − (ϕν(1 − σ) + 1)(1 − ν)(1 − σ)Ct (ct+1 , `t+2 ) x3t =(ν(1 − σ) − 1)ct − ϕν(1 − σ)ct−1 + (1 − ν)(1 − σ)`t x4t =ν(1 − σ)Et ct+1 − (ϕν(1 − σ) + 1)ct + (1 − ν)(1 − σ)Et `t+1 +
(ν(1 − σ))2 ((1 − ν)(1 − σ))2 Vt ct+1 + Vt `t+1 + ν(1 − ν)(1 − σ)2 Ct (ct+1 , `t+1 ) 2 2
where Et , Vt , Ct denote respectively conditional expectation, conditional variance and conditional covariance. Note that the specification of utility nests several types of preferences commonly considered in the literature. We focus on four sub–cases. The first one (SEP) constitutes the benchmark case and corresponds to the standard time–separable utility function with separability also between consumption and leisure. It is obtained by setting ν = 1 and ϕ = 0. The second one (SEP+HAB) relaxes the time–separability assumption by assuming ϕ > 0, and was too considered by Canzoneri 2
We consider habits a la Fuhrer, 2000. We have also derived the implications of the alternative specifications of habits studied by Canzoneri et al., 2006, without finding any improvement in the results. 3 Note that is is straightforward to derive the real interest rate from this equation.
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et al., 2006. The other two cases involve no-separable consumption–leisure. In the third case (NonSEP), we set ν ∈ (0, 1) while maintaining the time–separability of consumption decisions (ϕ = 0). In the forth case (NonSEP+HAB) we allow also for habits (ϕ > 0). In order to gain some understanding of how the presence of non-separability between consumption and leisure affects the properties of the model, we report below the Euler equations with and without separability in the absence of habits. SEP : (1 + it )−1 = β exp [−σEt γt+1 − Et πt+1 + Ωs )]
(1)
where γt ≡ log(Ct /Ct−1 ) and Ωs contains various —constant— second order moments N onSEP : (1 + it )−1 = β exp [(ν(1 − σ) − 1)Et γt+1 + (1 − ν)(1 − σ)Et ηt+1 − Et πt+1 + Ωn ] (2) where ηt ≡ log(`t /`t−1 ) and Ωn contains various —constant— second order moments. Equation 1 indicates that the relationship between consumption growth and real interest rates is always positive4 . As argued in the introduction, this is the reason why the Euler equation cannot capture the effects of changes in monetary policy in models with nominal rigidities. Equation 2 shows that this relationship is still positive but can become weaker if ν < 1. The sign of the relationship between leisure growth and the real interest rate is given by the sign of −(1 − ν)(1 − σ). Now consider an act of monetary tightening. As our VAR below establishes, expected consumption growth declines (as in Christiano et al., 2005), pushing the Euler real equation down, while expected leisure growth increases (expected employment growth declines). If σ > 1 then expected leisure growth pushes the real interest rate upwards. If the expected leisure growth effect dominates over the expected consumption growth one, then the real interest rate may increase following tighter monetary policy, an implication consistent with the data. Whether this happens or not depends not only on the coefficients of consumption and leisure growth but also on the coefficients of the various conditional second moments that enter the Euler equation. Nonetheless, it is worth noting that in the VAR described below, the effect of monetary policy on leisure is considerably greater than that on consumption growth (+0.04% vs −0.02%), which increases the likelihood that the Euler real interest rate will increase following a monetary contraction.
1.2
Parametrization and Empirical Specification
In order to derive the Euler implied interest rate we need information on the parameters as well as on the conditional moments appearing in the Euler equation. The parameter values are set as follows. The discount factor is set to β = 0.9926 so that the households discount the future at an annual rate of 3%. The risk aversion coefficient, σ, is set to 2 for time separable specifications and 4 The sign of the relationship between the real interest rate and consumption growth is determined by the sign of σ, which is always positive.
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6.11 for the habit persistence specification.5 The habit persistence parameter is set to ϕ = 0.8. In the case of non–separability between consumption and leisure, we set ν = 0.34, which implies a work share of 30% in the model without habits. We compute the conditional moments on consumption, leisure and the inflation rate using a VAR specification6 as suggested by Canzoneri et al. A(L)Yt = A0 + ut where A(L) = I −
Pp
i i=1 Ai L .
Yt consists of the log of real consumption expenditures in services
and non durable goods, ct , the inflation rate, πt , a measure of leisure, `t , the log of the CRB price index, crbt , the log of real disposable income, rdit , the log of non consumption expenditures, ymct , and the federal fund rate, f f rt .7 We consider a specification in levels8 and in first differences. Yt = {ct , πt , `t , crbt , rdit , ymct , f f rt }
or
Yt = {∆ct , πt , ∆`t , ∆crbt , ∆rdit , ∆ymct , f f rt }
The data are from the US over 1960:I to 2002:IV. Standard lag selection criteria favored a VAR(3) specification. The conditional second moments are constant and are given by the (i, j) elements of Vt (Yt+1 ) = Σ, Vt (Yt+2 ) = A1 ΣA01 + Σ, Ct (Yt+1 , Yt+2 ) = ΣA01 , where Σ is the estimated variance covariance matrix of ut .
2
The results
The results are virtually identical under either specification of the VAR (level or difference) so we report only those corresponding to the level specification. We derive the properties of the Euler implied nominal and real interest rates under the four alternative specifications of preferences. Table 1 reports the properties of these rates, Figure 1 compares the paths of model implied and actual rates and Figure 2 reports the response of implied and actual rates to a FFR shock. Three noteworthy patterns emerge. First, allowing for non–separability makes the correlation between actual and Euler implied rates strongly positive (in particular for the nominal rates, it is close to 0.6). The match of volatilities also gets tighter in comparison to the separable case (in particular for the specification with habits); see Table 1. As expected (from the equity and interest rate puzzle literature) all models miss the average level of interest rates, but this is not a fatal flaw for models of monetary policy. Second, the discrepancy in the paths of the two rates is lower. For instance, comparing the first and the third panels in Figure 1 it can be seen that the nonSEP specification captures much better interest rates movements: For instance, it can account for the large spike in 5
For comparability purposes we use the same values as Canzoneri et al., 2006. Of course one could choose a more complicated specification. But there is nothing in the model to justify this. And other choices might expose the analysis to data mining type of criticisms. 7 Further details on the variables can be found in the appendix. 8 Lowercases denote logs of variables. 6
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the mid 70s, the spike around 1990 as well as for some of the fast decline in the early 90s. And third, the response of the Euler equation rate to a FFR shock is now slightly (but insignificantly) positive. Of course, the model implied path is quantitatively still very far from that observed in the data, so non-separability is not sufficient on its own to resolve all of the problems afflicting the consumption Euler equation. Nonetheless, it seems fair to say that non–separability significantly improves the performance of the standard model relative to the separable case, in particular along the dimension of the correlation and dynamics. Table 1: Selected Moments
Data
SEP
SEP+HAB
NonSEP
NonSEP+HAB
Real Interest Rate Mean Std Corr
2.25 2.37 –
6.73 2.15 0.08
6.88 32.17 -0.03
5.46 1.21 0.27
5.03 2.35 0.22
Nominal Interest Rate Mean Std Corr
6.31 3.02 –
10.89 1.77 0.23
11.03 32.25 0.04
9.61 1.75 0.64
9.17 2.51 0.53
Note: SEP = Separable utility between consumption and leisure; SEP+HAB = Separable utility between consumption and leisure but with Fuhrer Habits; NonSEP= Consumption and leisure are separable; NonSEP+HAB =Non separable consumption and leisure and Fuhrer Habits.
References Abel, A., 1990, Asset Prices under Habit Formation and Catching Up with the Joneses, American Economic Review 80, no. 2, May, 38-42. Campbell, J. and Cochrane, J., 1999, By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior, Journal of Political Economy 107, no. 2, April, 205-251. Canzoneri, M., Cumby, R. and Diba, E., 2006, Euler Equations and Money Market Interest Rates: A Challenge for Monetary Policy Models, mimeo. Christiano, L., Eichenbaum, M. and Evans, C., 2005, Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy, Journal of Political Economy 13(1), 1-45. Fuhrer, J., 2000, Habit Formation in Consumption and Its Implications for Monetary-Policy Models, American Economic Review 90, no. 3, June, 367-390.
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Data Appendix The VAR includes the following variables: Real Consumption, c: Consumption expenditures in services and non durable goods, chained dollars 2000. (PCESVC96 and PCNDGC96), Ct =
PCESVC96+PCNDGC96 . CNP16OV
Real disposable income per capita,rdi: It is computed using real income (DPIC96) and population (CNP16OV, Civilian Population over 16): RDIt =
DPIC96 CNP16OV .
Federal Fund Rate,f f r. Inflation rate, π: Computed from nominal (PCESV and PCNDG) and real consumption (PCESVC96 and PCNDGC96) as Pct =
PCESV+PCNDG PCESVC96+PCNDGC96
Output less consumption, ymc: Y M Ct =
so that Π =
Pct Pct−1 .
GDPC1−(PCESVC96+PCNDGC96) . CNP16OV
Price of raw materials, crb: We use the CRB price index. Leisure: We use hours worked in the non farm business sector. Weekly hours are multiplied by the employment rate and then demeaned and re-scaled such that their mean is 1/3 (consistent with the standard model). Leisure, `t , is given by (1 − ht ).
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Figure 1: Data vs Model (a) SEP Annualized Nominal Interest Rate
Annualized Real Interest Rate
20
15 Model FFR
15
10
10
5
5
0
0
1970
1980
1990
−5
2000
1970
1980
1990
2000
(b) SEP+HAB Annualized Nominal Interest Rate
Annualized Real Interest Rate
150
150 Model FFR
100
100
50
50
0
0
−50
−50
−100
1970
1980
1990
−100
2000
1970
1980
1990
2000
(c) NonSEP Annualized Nominal Interest Rate
Annualized Real Interest Rate
20
15 Model FFR
15
10
10
5
5
0
0
1970
1980
1990
−5
2000
1970
1980
1990
2000
(d) NonSEP+HAB Annualized Nominal Interest Rate
Annualized Real Interest Rate
20
15 Model FFR
15
10
10
5
5
0
0
1970
1980
1990
−5
2000
9
1970
1980
1990
2000
Figure 2: Impulse Response Functions (FFR shock) (a) SEP −4
x10
Nominal Interest Rate
−4 x10 Euler Nominal Interest Rate
5
15
0
10
−5 5 −10 0
−15 5 −4
x10
10 15 Quarters Real Interest Rate
20
5
−4
x10
10 15 Quarters Euler Real Interest Rate
20
5
15
0
10
−5 5 −10 0 5
10 Quarters
15
−15
20
5
10 Quarters
15
20
(b) NonSEP+HAB −4
x10
Nominal Interest Rate
−3
x10
15
Euler Nominal Interest Rate
1
10
0
5 −1 0 5 −4
x10
10 15 Quarters Real Interest Rate
20
5
−3
x10
15
10 15 Quarters Euler Real Interest Rate
20
1
10
0
5 −1 0 5
10 Quarters
15
20
5
10 Quarters
15
20
Note: Shaded area: 95% confidence intervals obtained by bootstrap (1000 replications.
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